Vector Error Correction Model (VECM)

[mickeykozzi]

Good day to you all.

My name is Michael and I am doing a multivariable time series analysis for my PhD.
I have followed the usual steps (Please see the attached file below).

While watching instruction on how to run VECMs on youtube (from strong sources), I can see examples where after lag selection, Johansen's testing for cointegration requires to test for p-1 lags and lag selection for the VECM test also requires p-1. These authors claim that because the VECM model is rewriting the VAR by differencing and losing one lag, we must follow p-1 for lag selection. That is, if the VAR process, at level, suggests 3 lags for annual data, we use 2 lags for our Johansen's method and VECM testing (3-1) = 2 lags.
Please see below some examples of using EVIEWs and doing this:

https://www.youtube.com/watch?v=_5E77hTSjo4
https://www.youtube.com/watch?v=-5HjmawPq_w

Also, I have seen throughout instruction videos, that if Johansen's method suggests more than 1 cointegrating Vector, say 2, 3, 4, we can use 1 cointegrating vector within our VECM model. We can do this to reduce the complexity of results.
Please see below some examples of using EVIEWs and doing this:

https://www.youtube.com/watch?v=7BNbaGgmUf0
https://www.youtube.com/watch?v=l-0DgBzdKLQ

My questions

I have spoken to my PhD supervisor about this and she says it not always appropriate to follow online EVIEWS instructions, rather, she would like to see peer-reviewed journals and published books that suggest p-1 lags and only using 1 cointegrating vector is appropriate.

Does anyone have any experience or have read any peer-reviewed research or textbooks to find sources to reference? I am finding this task very difficult.

Thank you!

[EViews Gareth]

Without commenting on the particular question, I will point out that the first video you posted is from the IMF, and written by Sam Ouliaris which does give it considerable credibility.

[EViews Mirza]

Why do you need peer-reviewed journals to show this? You can do so yourself directly. It's pretty rudimentary.
I'll give you the case for p = 1 and p = 2. You can use mathematical induction or a brute force method to demonstrate the general case.

VAR(1) => Y_t = Gamma * Y_{t-1} + Beta * X + eps_t
= D(Y_t) +Y_{t-1}
= Gamma * Y_{t-1} + Beta * X + eps_t
Thus
D(Y_t) = Gamma * Y_{t-1} - Y_{t-1} + Beta * X + eps_t
= -(I - Gamma) * Y_{t-1} + Beta * X + eps_t
= Pi * Y_{t - 1} + Beta * X + eps_t
<= which is a VEC(0) model

VAR(2) => Y_t = Gamma_1 * Y_{t-1} + Gamma_2 * Y_{t-2} + Beta * X + eps_t
= D(Y_t) +Y_{t-1}
= Gamma_1 * Y_{t-1} + Gamma_2 * (- D(Y_{t-1}) + Y_{t-1}) + Beta * X + eps_t
Thus
D(Y_t) = Gamma * Y_{t-1} - Y_{t-1} + Gamma_2 * Y_{t-1} - Gamma_2 * D(Y_{t-1}) - Beta * X + eps_t
= – (I - Gamma_1 - Gamma_2) * Y_{t-1} - Gamma_2 * D(Y_{t-1}) + Beta * X + eps_t
= Pi * Y_{t - 1} - Gamma_2 * D(Y_{t-1}) + Beta * X + eps_t
<= which is a VEC(1) model

Clearly, a VAR(p) model has an equivalent representation as a VEC(p-1) model. In other words, any lag testing on the VEC model will be done one one lag less than the equivalent VAR model.

As for the number of cointegrating relationships to use, there is nothing in the literature that suggests one should use only one cointegrating relationship when one has several to choose from. Choosing only one is simply a convenience that reduces model complexity and interpretation. Hope this helps.

[mickeykozzi]

Mirza

Thank you for your explanation.
Would you also then conclude that (p-1) for the Johansen's test is appropriate as well?

Thank you.

[EViews Mirza]

Yes. The Johansen test is done on the VECM representation.

[mickeykozzi]

Mirza,

Thank you for replying once again!
Just so I understand, because a Var(p) is an equivalent representation of a VECM (p-1), we must also use (p-1) when conducting a Johansens test for cointegration as we intend to use VECM if cointegration is found using this test. For example, if the lag selection criteria suggest 3 lags for annual data, we use 2 lags 1 2) for our Johansen test and (1 2) for our VECM.

In regards to using 1 cointegrating vector when Johansen's may suggest 2 or more, I am testing specific model equations.
In these equations, I am adding extra variables to see if they improve the model.

For example, we may have a model, through economic theory that states:

Yt = A + B

I want to include 2 new variables, C and D

Yt = A + B + C + D

If Johansen's suggests there are 2 cointegrating vectors, I would need to place restrictions on the mode, that is have A or B = 0, or C or D = 0. This would go against my orginal aim, to see if the inclusion of variables C and D improve the model.

Hence why I wanted to only use 1 cointegrating vector to see if any short and long run relationships exist.

Does this make sense?

Thanks!!